Coroll. I. If the strait line A F be drawn, it will be perpendicular to the strait line E K. For both the angles at E will be equal, by reason of the equality of the two angles A B E and F B E, and of the two sides A B and F B. Coroll. II. If upon any point between B and D there fall a strait line, as A C, whose reflected line is C H, this also produced beyond C, will fall upon F; which is evident by the demonstration above. [Sidenote: If two strait parallel lines, drawn not oppositely, but from the same parts, fall upon the circumference of a circle, the lines reflected from them, if produced they meet within the circle, will make an angle double to that which is made by two strait lines drawn from the centre to the points of incidence.] 3. If from two points taken without a circle, two strait parallel lines, drawn not oppositely, but from the same parts, fall upon the circumference; the lines reflected from them, if produced they meet within the circle, will make an angle double to that which is made by two strait lines drawn from the centre to the points of incidence. Let the two strait parallels A B and D C (in fig. 3) fall upon the circumference B C at the points B and C; and let the centre of the circle be E; and let A B reflected be B F, and D C reflected be C G; and let the lines F B and G C produced meet within the circle in H; and let E B and E C be connected. I say the angle F H G is double to the angle B E C. For seeing A B and D C are parallels, and E B cuts A B in B, the same E B produced will cut D C somewhere; let it cut it in D; and let D C be produced howsoever to I, and let the intersection of D C and B F be at K. The angle therefore I C H, being external to the triangle C K H, will be equal to the two opposite angles C K H and C H K. Again, I C E being external to the triangle C D E, is equal to the two angles at D and E. Wherefore the angle I C H, being double to the angle I C E, is equal to the angles at D and E twice taken; and therefore the two angles C K H and C H K are equal to the two angles at D and E twice taken. But the angle C K H is equal to the angles D and A B D, that is, D twice taken; for A B and D C being parallels, the altern angles D and A B D are equal. Wherefore C H K, that is the angle F H G is also equal to the angle at E twice taken; which was to be proved. Coroll. If from two points taken within a circle two strait parallels fall upon the circumference, the lines reflected from them shall meet in an angle, double to that which is made by two strait lines drawn from the centre to the points of incidence. For the parallels A B and I C falling upon the points B and C, are reflected in the lines B H and C H, and make the angle at H double to the angle at E, as was but now demonstrated.